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Continuity and Differentiability

Short Answer Type

301. Differentiate the following w.r.t.x:

fraction numerator straight e to the power of straight x over denominator 1 plus cos space straight x end fraction

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302. Differentiate the following w.r.t.x:

straight e to the power of sin square root of straight x end exponent

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303. Differentiate the following w.r.t.x:

straight e to the power of 3 straight x end exponent log left parenthesis sin space 2 straight x right parenthesis

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304. Differentiate the following w.r.t.x:

straight e to the power of square root of 1 minus straight x squared end root end exponent

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305.

Differentiate space straight e to the power of straight x space log square root of straight x. tan space straight x space straight w. straight r. straight t. straight x.

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306. Differentiate the following functions w.r.t.x:

sin left parenthesis log space straight x right parenthesis

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307. Differentiate the following functions w.r.t.x:

straight e to the power of cos space straight x end exponent

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308. Differentiate the following functions w.r.t.x:

log left parenthesis cos space 5 straight x right parenthesis

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309. Differentiate the following functions w.r.t.x:

log space sin left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis

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310. Differentiate the following functions w.r.t.x: $log space tan open parentheses straight pi over 4 plus straight x over 2 close parentheses$

Let space space space space space straight y equals log space tan open parentheses straight pi over 4 plus straight x over 2 close parentheses therefore space dy over dx equals fraction numerator 1 over denominator open parentheses straight pi over 4 plus straight x over 2 close parentheses end fraction. straight d over dx open square brackets tan open parentheses straight pi over 4 plus straight x over 2 close parentheses close square brackets space space space space space space space space space space space space space equals fraction numerator 1 over denominator tan open parentheses straight pi over 4 plus straight x over 2 close parentheses end fraction. sec squared open parentheses straight pi over 4 plus straight x over 2 close parentheses. straight d over dx open parentheses straight pi over 4 plus straight x over 2 close parentheses space space space space space space space space space space space space space equals fraction numerator cos open parentheses straight pi over 4 plus straight x over 2 close parentheses over denominator sin open parentheses straight pi over 4 plus straight x over 2 close parentheses end fraction. fraction numerator 1 over denominator cos squared open parentheses straight pi over 4 plus straight x over 2 close parentheses end fraction.1 half space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 sin open parentheses straight pi over 4 plus straight x over 2 close parentheses cos open parentheses straight pi over 4 plus straight x over 2 close parentheses end fraction equals fraction numerator 1 over denominator sin open parentheses straight pi over 2 plus straight x close parentheses end fraction equals fraction numerator 1 over denominator cos space straight x end fraction equals sec space straight x

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